Generalizedqanalysis of log-periodicity: Applications to critical ruptures

We introduce a generalization of the q analysis, which provides a nonparametric tool for the description and detection of log-periodic structures associated with discrete scale invariance. We use this generalized q analysis to construct a signature called the $\mathrm{}(H,q)\mathrm{}$ derivative of discrete scale invariance, which we use to detect the log-periodicity in the cumulative energy release preceding the rupture of five pressure tanks made of composite carbon-matrix material. We investigate the significance level of the spectral Lomb periodogram of the optimal $\mathrm{}(H,q)\mathrm{}$ derivative. We confirm and strengthen previous parametric results that the cumulative energy release exhibits log-periodicity before rupture. However, our tests to use this method as a scheme for the prediction of the critical value of the stress at rupture are not encouraging.